Proof of (c). The proof can be found in [1]}. We give a simple version for the sake of completeness. If \( \gamma =0 \) , then we must have \( \alpha \lambda _j = 1 \)  for all \( 1\le j\le p \)  under the assumed conditions. One can verify that \( \Delta \)  will be a zero matrix in this case. Consequently, the eigenvalues of \( D \)  are all zero, i.e., \( \rho (D)=0 \) . We then consider the case of \( \gamma >0 \) . First, note that when \( \gamma >0 \) , \( D \)  is nonsingular, i.e., \( D \)  has no zero eigenvalue. Then we construct a matrix function of \( \lambda \)  as
\(G(\lambda ) = \begin{bmatrix}1+\gamma - \alpha \lambda & -\gamma \\1&0\end{bmatrix} \in \mathbb {R}^{2\times 2}.\) 
